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Theorem neipeltop 20843
Description: Lemma for neiptopreu 20847. (Contributed by Thierry Arnoux, 6-Jan-2018.)
Hypothesis
Ref Expression
neiptop.o 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
Assertion
Ref Expression
neipeltop (𝐶𝐽 ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
Distinct variable groups:   𝑝,𝑎,𝐶   𝑁,𝑎   𝑋,𝑎
Allowed substitution hints:   𝐽(𝑝,𝑎)   𝑁(𝑝)   𝑋(𝑝)

Proof of Theorem neipeltop
StepHypRef Expression
1 eleq1 2686 . . . 4 (𝑎 = 𝐶 → (𝑎 ∈ (𝑁𝑝) ↔ 𝐶 ∈ (𝑁𝑝)))
21raleqbi1dv 3135 . . 3 (𝑎 = 𝐶 → (∀𝑝𝑎 𝑎 ∈ (𝑁𝑝) ↔ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
3 neiptop.o . . 3 𝐽 = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝𝑎 𝑎 ∈ (𝑁𝑝)}
42, 3elrab2 3348 . 2 (𝐶𝐽 ↔ (𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
5 0ex 4750 . . . . . . 7 ∅ ∈ V
6 eleq1 2686 . . . . . . 7 (𝐶 = ∅ → (𝐶 ∈ V ↔ ∅ ∈ V))
75, 6mpbiri 248 . . . . . 6 (𝐶 = ∅ → 𝐶 ∈ V)
87adantl 482 . . . . 5 ((∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) ∧ 𝐶 = ∅) → 𝐶 ∈ V)
9 elex 3198 . . . . . . 7 (𝐶 ∈ (𝑁𝑝) → 𝐶 ∈ V)
109ralimi 2947 . . . . . 6 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → ∀𝑝𝐶 𝐶 ∈ V)
11 r19.3rzv 4036 . . . . . . 7 (𝐶 ≠ ∅ → (𝐶 ∈ V ↔ ∀𝑝𝐶 𝐶 ∈ V))
1211biimparc 504 . . . . . 6 ((∀𝑝𝐶 𝐶 ∈ V ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V)
1310, 12sylan 488 . . . . 5 ((∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) ∧ 𝐶 ≠ ∅) → 𝐶 ∈ V)
148, 13pm2.61dane 2877 . . . 4 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → 𝐶 ∈ V)
15 elpwg 4138 . . . 4 (𝐶 ∈ V → (𝐶 ∈ 𝒫 𝑋𝐶𝑋))
1614, 15syl 17 . . 3 (∀𝑝𝐶 𝐶 ∈ (𝑁𝑝) → (𝐶 ∈ 𝒫 𝑋𝐶𝑋))
1716pm5.32ri 669 . 2 ((𝐶 ∈ 𝒫 𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)) ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
184, 17bitri 264 1 (𝐶𝐽 ↔ (𝐶𝑋 ∧ ∀𝑝𝐶 𝐶 ∈ (𝑁𝑝)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  {crab 2911  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-in 3562  df-ss 3569  df-nul 3892  df-pw 4132
This theorem is referenced by:  neiptopuni  20844  neiptoptop  20845  neiptopnei  20846  neiptopreu  20847
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